Juliá was born at the Floral Park subsection of San Juan, the son of Olga Arcelay and Raúl Juliá. He was the oldest of four brothers and sisters; his brother Rafael died in an automobile accident in 1960. His mother was a mezzo-soprano who abandoned a potential career as a singer when she married Juliá's father. Some relatives on his father's side were part-time musicians.

Raúl's father was the founder of "La Cueva del Chicken Inn", a restaurant in San Juan. It was modeled after a similar restaurant in Madrid, Spain, called "Las Cuevas de Luis Candelas". Raúl's father dreamed of bringing pizza to Puerto Rico, and made that dream a reality when he hired an Italian cook in New York City who could prepare authentic Italian pizza for Puerto Rican palates. Raúl's sister María Eugenia claims that their father was, in a way, the first fast food mogul Puerto Rico ever had, since the relatively simple food would ensure prompt service at the restaurant. He founded the restaurant at the very house where Raúl and his brother Rafael were born, the brothers and sisters literally grew up with the family business, and the property is still owned by the Juliá family.